Problem: Sum
of Unit Fractions (click here to see the problem statement)
My Solution.
We are looking for
where
,
,
and ![](../UnitFractions/image04.gif)
Simplifying, ![](../UnitFractions/image05.gif)
We have, ![](../UnitFractions/image06.gif)
If we let
,
then ![](../UnitFractions/image08.gif)
So,![](../UnitFractions/image09.gif)
Which implies that :
![](../UnitFractions/image10.gif)
Which can only be true when
,
so ![](../UnitFractions/image12.gif)
Using the relationship
,
I set up a table
![](../UnitFractions/image14.gif)
![](../UnitFractions/image15.gif)
So the answer to the question,
"Which of these fractions (
, where n is a positive integer)
may be expressed as the sum of two different unit fractions of
integers?" appears to be: ALL OF THEM.